This is an orbifold Kähler metric: on the orbifold chart
(REF ) it pulls back to a genuine Kähler metric on \({n-1}\) . In fact it follows
from Lemma REF  below that it is the curvature of a hermitian
metric \(h_1\)  on the orbifold line
bundle \(\mathcal {O}_{\mathbb {P}(V)}(1)\) . The dual of this hermitian metric is the one of three natural
candidates
for the name of Fubini-Study metric on \(\mathcal {O}_{\mathbb {P}(V)}(-1)\) . A second natural choice \(h_2\) 
is given by \(|v|^2_{h_2} = \sum _{i} |v_{k+i}|^{\frac{2}{k+i}}\)  (note that \(|v|^2 = \sum _i|v_{k+i}|^2\) 
does not scale correctly under the action of \(*\)  to define a hermitian metric).
However it is the third candidate \(h_3=h_{FS}\)  below that we choose. It should be noted
that only on an unweighted projective space do all three agree and metrics. It seems that \(h_{FS}\)  is a special case of the more general metrics on
line bundles over toric varieties constructed by Batyrev-Tschinkel [1]}.
